Question

3 (3,7) over the line y = x
4 (-2, -8) over the x-axis
5 (-4, -5) over the y-axis
6 (0, -6) over the line y = x

Explanation:

To solve the problem of reflecting points over different lines (or axes), we use the following reflection rules:

Reflection over the line \( y = x \):

The rule for reflecting a point \( (a, b) \) over the line \( y = x \) is to swap the \( x \)-coordinate and the \( y \)-coordinate. So, the image of \( (a, b) \) is \( (b, a) \).

Reflection over the \( x \)-axis:

The rule for reflecting a point \( (a, b) \) over the \( x \)-axis is to keep the \( x \)-coordinate the same and change the sign of the \( y \)-coordinate. So, the image of \( (a, b) \) is \( (a, -b) \).

Reflection over the \( y \)-axis:

The rule for reflecting a point \( (a, b) \) over the \( y \)-axis is to change the sign of the \( x \)-coordinate and keep the \( y \)-coordinate the same. So, the image of \( (a, b) \) is \( (-a, b) \).

Now, let's solve each part:

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3. Reflect \( (3, 7) \) over the line \( y = x \)

Step 1: Apply the reflection rule for \( y = x \)

For a point \( (a, b) \), the reflection over \( y = x \) is \( (b, a) \). Here, \( a = 3 \) and \( b = 7 \).

So, the image is \( (7, 3) \).

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4. Reflect \( (-2, -8) \) over the \( x \)-axis

Step 1: Apply the reflection rule for the \( x \)-axis

For a point \( (a, b) \), the reflection over the \( x \)-axis is \( (a, -b) \). Here, \( a = -2 \) and \( b = -8 \).

So, \( -b = -(-8) = 8 \).

Thus, the image is \( (-2, 8) \).

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5. Reflect \( (-4, -5) \) over the \( y \)-axis

Step 1: Apply the reflection rule for the \( y \)-axis

For a point \( (a, b) \), the reflection over the \( y \)-axis is \( (-a, b) \). Here, \( a = -4 \) and \( b = -5 \).

So, \( -a = -(-4) = 4 \).

Thus, the image is \( (4, -5) \).

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6. Reflect \( (0, -6) \) over the line \( y = x \)

Step 1: Apply the reflection rule for \( y = x \)

For a point \( (a, b) \), the reflection over \( y = x \) is \( (b, a) \). Here, \( a = 0 \) and \( b = -6 \).

So, the image is \( (-6, 0) \).

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Final Answers:

1. \( \boldsymbol{(7, 3)} \)
2. \( \boldsymbol{(-2, 8)} \)
3. \( \boldsymbol{(4, -5)} \)
4. \( \boldsymbol{(-6, 0)} \)

Answer:

To solve the problem of reflecting points over different lines (or axes), we use the following reflection rules:

Reflection over the line \( y = x \):

The rule for reflecting a point \( (a, b) \) over the line \( y = x \) is to swap the \( x \)-coordinate and the \( y \)-coordinate. So, the image of \( (a, b) \) is \( (b, a) \).

Reflection over the \( x \)-axis:

The rule for reflecting a point \( (a, b) \) over the \( x \)-axis is to keep the \( x \)-coordinate the same and change the sign of the \( y \)-coordinate. So, the image of \( (a, b) \) is \( (a, -b) \).

Reflection over the \( y \)-axis:

The rule for reflecting a point \( (a, b) \) over the \( y \)-axis is to change the sign of the \( x \)-coordinate and keep the \( y \)-coordinate the same. So, the image of \( (a, b) \) is \( (-a, b) \).

Now, let's solve each part:

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3. Reflect \( (3, 7) \) over the line \( y = x \)

Step 1: Apply the reflection rule for \( y = x \)

For a point \( (a, b) \), the reflection over \( y = x \) is \( (b, a) \). Here, \( a = 3 \) and \( b = 7 \).

So, the image is \( (7, 3) \).

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4. Reflect \( (-2, -8) \) over the \( x \)-axis

Step 1: Apply the reflection rule for the \( x \)-axis

For a point \( (a, b) \), the reflection over the \( x \)-axis is \( (a, -b) \). Here, \( a = -2 \) and \( b = -8 \).

So, \( -b = -(-8) = 8 \).

Thus, the image is \( (-2, 8) \).

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5. Reflect \( (-4, -5) \) over the \( y \)-axis

Step 1: Apply the reflection rule for the \( y \)-axis

For a point \( (a, b) \), the reflection over the \( y \)-axis is \( (-a, b) \). Here, \( a = -4 \) and \( b = -5 \).

So, \( -a = -(-4) = 4 \).

Thus, the image is \( (4, -5) \).

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6. Reflect \( (0, -6) \) over the line \( y = x \)

Step 1: Apply the reflection rule for \( y = x \)

For a point \( (a, b) \), the reflection over \( y = x \) is \( (b, a) \). Here, \( a = 0 \) and \( b = -6 \).

So, the image is \( (-6, 0) \).

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Final Answers:

1. \( \boldsymbol{(7, 3)} \)
2. \( \boldsymbol{(-2, 8)} \)
3. \( \boldsymbol{(4, -5)} \)
4. \( \boldsymbol{(-6, 0)} \)